Continuous-time mean-variance efficiency: the 80% rule

This paper studies a continuous-time market where an agent, having specified an investment horizon and a targeted terminal mean return, seeks to minimize the variance of the return. The optimal portfolio of such a problem is called mean-variance efficient \`{a} la Markowitz. It is shown that, when the market coefficients are deterministic functions of time, a mean-variance efficient portfolio realizes the (discounted) targeted return on or before the terminal date with a probability greater than 0.8072. This number is universal irrespective of the market parameters, the targeted return and the length of the investment horizon.Comment: Published at http://dx.doi.org/10.1214/105051606000000349 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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The outer profile of dark matter halos: an analytical approach

A steepening feature in the outer density profiles of dark matter halos indicating the splashback radius has drawn much attention recently. Possible observational detections have even been made for galaxy clusters. Theoretically, Adhikari et al. have estimated the location of the splashback radius by computing the secondary infall trajectory of a dark matter shell through a growing dark matter halo with an NFW profile. However, since they imposed a shape of the halo profile rather than computing it consistently from the trajectories of the dark matter shells, they could not provide the full shape of the dark matter profile around the splashback radius. We improve on this by extending the self-similar spherical collapse model of Fillmore \& Goldreich to a $\Lambda$CDM universe. This allows us to compute the dark matter halo profile and the trajectories simultaneously from the mass accretion history. Our results on the splashback location agree qualitatively with Adhikari et al. but with small quantitative differences at large mass accretion rates. We present new fitting formulae for the splashback radius $R_{\rm sp}$ in various forms, including the ratios of $R_{\rm sp} / R_{\rm 200c}$ and $R_{\rm sp} / R_{\rm 200m}$. Numerical simulations have made the puzzling discovery that the splashback radius scales well with $R_{\rm 200m}$ but not with $R_{\rm 200c}$. We trace the origin of this to be the correlated increase of $\Omega_{\rm m}$ and the average halo mass accretion rate with an increasing redshift.Comment: 10 pages, 10 figures, published in MNRA

PTOMSM: A modified version of Topological Overlap Measure used for predicting Protein-Protein Interaction Network

A variety of methods are developed to integrating diverse biological data to predict novel interaction relationship between proteins. However, traditional integration can only generate protein interaction pairs within existing relationships. Therefore, we propose a modified version of Topological Overlap Measure to identify not only extant direct PPIs links, but also novel protein interactions that can be indirectly inferred from various relationships between proteins. Our method is more powerful than a naïve Bayesian-network-based integration in PPI prediction, and could generate more reliable candidate PPIs. Furthermore, we examined the influence of the sizes of training and test datasets on prediction, and further demonstrated the effectiveness of PTOMSM in predicting PPI. More importantly, this method can be extended naturally to predict other types of biological networks, and may be combined with Bayesian method to further improve the prediction